Identifier
Values
=>
Cc0020;cc-rep-0 Cc0029;cc-rep
([],1)=>([],1)=>0 ([],2)=>([],1)=>0 ([(0,1)],2)=>([(0,1)],2)=>1 ([],3)=>([],1)=>0 ([(1,2)],3)=>([(0,1)],2)=>1 ([(0,2),(1,2)],3)=>([(0,1),(0,2),(1,3),(2,3)],4)=>2 ([(0,1),(0,2),(1,2)],3)=>([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)=>1 ([],4)=>([],1)=>0 ([(2,3)],4)=>([(0,1)],2)=>1 ([(1,3),(2,3)],4)=>([(0,1),(0,2),(1,3),(2,3)],4)=>2 ([(0,3),(1,3),(2,3)],4)=>([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)=>3 ([(0,3),(1,2)],4)=>([(0,1),(0,2),(1,3),(2,3)],4)=>2 ([(0,3),(1,2),(2,3)],4)=>([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)=>3 ([(1,2),(1,3),(2,3)],4)=>([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)=>1 ([],5)=>([],1)=>0 ([(3,4)],5)=>([(0,1)],2)=>1 ([(2,4),(3,4)],5)=>([(0,1),(0,2),(1,3),(2,3)],4)=>2 ([(1,4),(2,4),(3,4)],5)=>([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)=>3 ([(1,4),(2,3)],5)=>([(0,1),(0,2),(1,3),(2,3)],4)=>2 ([(1,4),(2,3),(3,4)],5)=>([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)=>3 ([(0,1),(2,4),(3,4)],5)=>([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)=>3 ([(2,3),(2,4),(3,4)],5)=>([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)=>1 ([],6)=>([],1)=>0 ([(4,5)],6)=>([(0,1)],2)=>1 ([(3,5),(4,5)],6)=>([(0,1),(0,2),(1,3),(2,3)],4)=>2 ([(2,5),(3,5),(4,5)],6)=>([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)=>3 ([(2,5),(3,4)],6)=>([(0,1),(0,2),(1,3),(2,3)],4)=>2 ([(2,5),(3,4),(4,5)],6)=>([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)=>3 ([(1,2),(3,5),(4,5)],6)=>([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)=>3 ([(3,4),(3,5),(4,5)],6)=>([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)=>1 ([(0,5),(1,4),(2,3)],6)=>([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)=>3 ([],7)=>([],1)=>0 ([(5,6)],7)=>([(0,1)],2)=>1 ([(4,6),(5,6)],7)=>([(0,1),(0,2),(1,3),(2,3)],4)=>2 ([(3,6),(4,6),(5,6)],7)=>([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)=>3 ([(3,6),(4,5)],7)=>([(0,1),(0,2),(1,3),(2,3)],4)=>2 ([(3,6),(4,5),(5,6)],7)=>([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)=>3 ([(2,3),(4,6),(5,6)],7)=>([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)=>3 ([(4,5),(4,6),(5,6)],7)=>([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)=>1 ([(1,6),(2,5),(3,4)],7)=>([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)=>3
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Description
The dimension of the space of valuations of a lattice.
A valuation, or modular function, on a lattice $L$ is a function $v:L\mapsto\mathbb R$ satisfying
$$ v(a\vee b) + v(a\wedge b) = v(a) + v(b). $$
It was shown by Birkhoff [1, thm. X.2], that a lattice with a positive valuation must be modular. This was sharpened by Fleischer and Traynor [2, thm. 1], which states that the modular functions on an arbitrary lattice are in bijection with the modular functions on its modular quotient Mp00196The modular quotient of a lattice..
Moreover, Birkhoff [1, thm. X.2] showed that the dimension of the space of modular functions equals the number of subsets of projective prime intervals.
Map
connected vertex partitions
Description
Sends a graph to the lattice of its connected vertex partitions.
A connected vertex partition of a graph $G = (V,E)$ is a set partition of $V$ such that each part induced a connected subgraph of $G$. The connected vertex partitions of $G$ form a lattice under refinement. If $G = K_n$ is a complete graph, the resulting lattice is the lattice of set partitions on $n$ elements.
In the language of matroid theory, this map sends a graph to the lattice of flats of its graphic matroid. The resulting lattice is a geometric lattice, i.e. it is atomistic and semimodular.